General Solution for Cosine (negative angles)

Question

• $cos(2(x+\pi/3))=1/2$
• $2(x+\pi/3)=\pi/3$
• $x+\pi/3=\pi/6$
• $x+2\pi/6=\pi/6$
• $x=-\pi/6$
• $x=5\pi/6$ (is this step correct) ... ??
• $x = +/- \pi/6 +k\pi$ , $k$ is a subset of $Z$
• $x = +/- 5\pi/6 +k\pi$ , $k$ is a subset of $Z$

can someone please look over my working out, and correct me

Answer 1

$$\cos2\left(x+\frac\pi3\right)=\frac12=\cos\frac\pi3$$

$$\implies2\left(x+\frac\pi3\right)=2m\pi\pm\frac\pi3$$ where $m$ is any integer

$$\implies x+\frac\pi3=m\pi\pm\frac\pi6$$

$\displaystyle'+'\implies x+\frac\pi3=m\pi+\frac\pi6\iff x=m\pi-\frac\pi6$

etc.